Problem: $\int ( x^4 + x +9)\,dx=$ $+C$
Answer: We can use the sum rule and the constant multiple rule for indefinite integrals: $\begin{aligned} &\int [f(x)+g(x)]dx=\int f(x)\,dx+\int g(x)\,dx \\\\\\ &\int k\cdot f(x)= k\cdot\int f(x)\,dx \end{aligned}$ Using the sum and the constant multiple rules, we can rewrite our integral as follows: $\int ( x^4 + x +9)\,dx= \int x^4\,dx +\int x\,dx +9\int 1\,dx$ Now we can find each indefinite integral using the reverse power rule: $\int x^n\,dx=\dfrac{x^{n+1}}{n+1}+C$ Note: we can only use the reverse power rule because $n \neq -1$. $\begin{aligned} &\phantom{=}\int ( x^4 + x +9)\,dx \\\\ &= \int x^4\,dx +\int x\,dx +9\int 1\,dx \\\\ &= \dfrac{x^5}{5} +\dfrac{x^2}{2} +9\dfrac{x^1}{1}+C \\\\ &=\dfrac{1}{5} x^5 +\dfrac{1}{2} x^2 +9 x+C \end{aligned}$ In conclusion, $\int ( x^4 + x +9)\,dx=\dfrac{1}{5} x^5 +\dfrac{1}{2} x^2 +9 x+C$